Cohomological Hall Algebras of Calabi-Yau 3-folds

“卡-丘3-折叠”是6维弯曲空间，在几何学中很重要，在理论物理学的弦理论中也很重要。弦论始终只在10维时空中定义，而为了描述我们的4维时空（3个空间维度和1个时间维度），我们需要将额外的6维包裹在一个非常小的卡-丘3重上。卡丘三重的几何决定了我们的四维物理学（粒子等）。通过将四维物理理论与卡-丘3重联系起来，弦论家们对卡-丘3重做出了惊人的解释，这是一个被称为镜像对称的领域。“唐纳森-托马斯不变量”是对生活在卡-丘3重X上的几何对象（相干层）进行计数的数字。相干层形成一个“模空间”M，一个奇异空间，DT不变量由M上一种不寻常的积分定义。在弦论中，DT不变量是“BPS状态的数量”，一种粒子。在2008年，PI和Yinan Song展示了如何在最一般的情况下定义DT不变量，并证明了当X上的结构变形时，它们会通过“跨壁公式”发生变化。这导致了对唐纳森-托马斯理论及其扩展的研究爆炸。2013年，PI证明DT不变量可以解释为向量空间的维数。作为模空间M的一种奇异上同调，向量空间的构造是困难的。（上同调测量空间M的“形状”，例如甜甜圈中的洞。在弦论中，这些向量空间是“BPS态的向量空间”，是与X相关的量子场论的一部分。（Kontsevich-Soibelman）和弦理论（哈维-摩尔），这些向量空间应该有一个乘法，使他们成为一个代数（加法和乘法的东西，如普通数）-物理学文献中的“BPS状态代数”，或数学文献中的“上同调霍尔代数（CoHA）”。在物理学中，倍增来自两个粒子结合成第三个粒子。这个猜想在2010年由Kontsevich-Soibelman证明为“超势的箭袋”，这是一个卡-丘3-折叠的玩具模型。PI和Pavel Safronov在2015年的工作使人们能够定义模空间M的小区域上的乘法，但还没有定义整个空间。在这个提议中，我们的目标是在BPS状态的向量空间上构造乘法。我们将通过一种新的方法证明2013年PI在移位辛导出代数几何领域的一个更一般的猜想来做到这一点。证明这个猜想使我们能够为Calabi-Yau 3-folds定义CoHA，这是一个无限维代数，DT不变量是这些代数的维数。然后我们可以用表示论来研究这些代数，例如，我们可以证明DT不变量是模形式的幂级数系数，模形式是数论中的一类特殊函数。我们将证明的猜想还有其他非常重要的应用，我们在提案中探索：* 它给出了Joyce和Borisov在2015年定义的8维Calabi-Yau 4-folds X的“DT 4不变量”的另一种构造，并表明这些DT 4不变量具有额外的有用性质，例如，它们在将X切割成两部分时的行为。它允许我们定义一个辛流形的“福谷范畴”的代数几何版本，这是镜像对称的关键。这种代数形式比通常的形式更简单、更严格，而且不需要许多通常的限制性假设。这个“福谷范畴”有重要的应用，包括研究普通三维空间中的纽结，以及将纽结的不变量转化为一种称为拓扑量子场论的数学结构。

"Calabi-Yau 3-folds" are 6-dimensional curved spaces that are important in Geometry, and also in String Theory in Theoretical Physics. String Theory is consistently defined only in 10-dimensional spacetime, and in order to describe our 4-dimensional spacetime (3 space dimensions and 1 time dimension), one is required to wrap the additional 6 dimensions on a very small Calabi-Yau 3-fold. The geometry of the Calabi-Yau 3-fold determines our 4-dimensional physics (particles, etc). By associating a 4-dimensional physical theory to the Calabi-Yau 3-fold, which is not mathematically understood, String Theorists make amazing conjectures about Calabi-Yau 3-folds, an area known as Mirror Symmetry."Donaldson-Thomas invariants" are numbers counting geometric objects (coherent sheaves) living on a Calabi-Yau 3-fold X. The coherent sheaves form a "moduli space" M, a singular space, and DT invariants are defined by an unusual kind of integration over M. In String Theory, DT invariants are "numbers of BPS states", a kind of particle. In 2008, the PI and Yinan Song showed how to define DT invariants in the most general case, and proved they change by a "wall-crossing formula" as the structure on X deforms. This led to an explosion of research on Donaldson-Thomas theory and its extensions. In 2013, the PI showed DT invariants can be interpreted as dimensions of vector spaces. The vector spaces have a difficult construction as a kind of exotic cohomology of the moduli space M. (Cohomology measures the "shape" of a space M, e.g. the hole in a donut.) In String Theory, these vector spaces are "vector spaces of BPS states", part of the Quantum Field Theory associated to X.It is a long standing conjecture in Geometry (Kontsevich-Soibelman) and String Theory (Harvey-Moore) that these vector spaces should have a multiplication on them making them into an algebra (something with addition and multiplication, like ordinary numbers) - the "algebra of BPS states" in the Physics literature, or "Cohomological Hall Algebra (CoHA)" in the mathematics literature. In Physics, the multiplication comes from two particles joining to make a third particle. The conjecture was proved in 2010 by Kontsevich-Soibelman for "quivers with superpotential", a toy model for Calabi-Yau 3-folds. Work by the PI and Pavel Safronov in 2015 enables one to define the multiplication over small regions of the moduli space M, but not yet over the whole space.In this proposal, we aim to construct the multiplication on the vector space of BPS states. We will do this by proving a much more general conjecture of the PI from 2013 in the area of Shifted Symplectic Derived Algebraic Geometry, by a new method.Proving this conjecture enables us to define CoHAs for Calabi-Yau 3-folds, which are infinite-dimensional algebras, and DT invariants are dimensions of pieces of these algebras. We can then study these algebras using Representation Theory, e.g. it may be possible to show that DT invariants are power series coefficients of modular forms, a class of special functions in Number Theory.The conjecture we will prove also has other very important applications, which we explore in the proposal:* It gives an alternative construction of "DT4 invariants" of 8-dimensional Calabi-Yau 4-folds X, defined by Joyce and Borisov in 2015, and shows these DT4 invariants have additional useful properties, e.g. how they behave on cutting X into 2 pieces.* It allows us to define an algebraic geometry version of the "Fukaya category" of a symplectic manifold, which are key to Mirror Symmetry. This algebraic version will be simpler and more rigid than the usual version, and work without many of the usual restrictive assumptions.* This "Fukaya category" has important applications, including to the study of knots in ordinary 3-dimensional space, and to making invariants of knots into a mathematical structure called a Topological Quantum Field Theory.